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ERIC Number: EJ1048209
Record Type: Journal
Publication Date: 2014-Dec
Pages: 26
Abstractor: As Provided
Reference Count: 24
ISSN: ISSN-1076-9986
Handling Correlations between Covariates and Random Slopes in Multilevel Models
Bates, Michael David; Castellano, Katherine E.; Rabe-Hesketh, Sophia; Skrondal, Anders
Journal of Educational and Behavioral Statistics, v39 n6 p524-549 Dec 2014
This article discusses estimation of multilevel/hierarchical linear models that include cluster-level random intercepts and random slopes. Viewing the models as structural, the random intercepts and slopes represent the effects of omitted cluster-level covariates that may be correlated with included covariates. The resulting correlations between random effects (intercepts and slopes) and included covariates, which we refer to as "cluster-level endogeneity," lead to bias when using standard random effects (RE) estimators such as (restricted) maximum likelihood. While the problem of correlations between unit-level covariates and random intercepts is well known and can be handled by fixed-effects (FE) estimators, the problem of correlations between unit-level covariates and random slopes is rarely considered. When applied to models with random slopes, the standard FE estimator does not rely on standard cluster-level exogeneity assumptions, but requires an "uncorrelated variance assumption" that the variances of unit-level covariates are uncorrelated with their random slopes. We propose a "per-cluster regression" (PC) estimator that is straightforward to implement in standard software, and we show analytically that it is unbiased for all regression coefficients under cluster-level endogeneity and violation of the uncorrelated variance assumption. The PC, RE, and an augmented FE estimator are applied to a real data set and evaluated in a simulation study that demonstrates that our PC estimator performs well in practice.
SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail:; Web site:
Publication Type: Journal Articles; Reports - Research
Education Level: N/A
Audience: N/A
Language: English
Sponsor: Institute of Education Sciences (ED)
Authoring Institution: N/A
IES Funded: Yes
Grant or Contract Numbers: R305B090011; R305B110017